A scheme for determining stepsizes for unconstrained optimization methods

We present a new scheme for determining stepsizes for iterative unconstrained minimization methods. This scheme provides a stepsize estimate for the efficient Armijo-type stepsize determination rule and improves its performance. As examples for the new scheme, we also present a new gradient algorithm and a new conjugate gradient algorithm. These two algorithms are readily implementable and eventually demand only one trial stepsize at each iteration. Their global convergence is established without any convexity assumptions. The convergence ratio associated with the gradient algorithm is shown to converge to the canonical convergence ratio (that is, the best possible convergence ratio). The convergence rate of the conjugate gradient algorithm is n-step superlinear and n-step quadratic.